{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h2>Bayesian Inference of the mean of a Gaussian</h2>\n",
    "<p>In this notebook, we will use Bayesian inference to infer the mean of a Gaussian. We assume that we observe $N$ values that have come from a Gaussian with mean $\\mu$ and variancs $\\sigma^2$. Assuming that we know $\\sigma^2=1$, we would like to determine $\\mu$.</p>\n",
    "\n",
    "<p>We start by defining a prior density on $\\mu$. We will use a Gaussian as our likelihood will be Gaussian (as per our assumption above) and the Gaussian prior is conjugate to the Gaussian likelihood. We (arbitrarily - in real problems you should think carefully about prior values) choose a prior with mean $a=0$ and variance $b^2=1$</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "true_mu = 2.0\n",
    "a = 0\n",
    "b_sq = 1\n",
    "sig_sq = 1\n",
    "x = np.random.normal(true_mu,np.sqrt(sig_sq),(10,1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p>The plot below shows our choice of prior density:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x10840fbd0>"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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79GbWYGZzzazJzK5v4Zx/N7N5ZjbTzA7Ne/1dM5tlZjPM7MXoH0Gqxj33wDHH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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10831ea10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import pylab as plt\n",
    "%matplotlib inline\n",
    "\n",
    "def normal_pdf(x,mu,sigma):\n",
    "    return (1.0/(sigma*np.sqrt(2*np.pi)))*np.exp(-(1.0/(2.0*sigma**2))*(x-mu)**2)\n",
    "\n",
    "plotx = np.arange(-5,5,0.01)\n",
    "plt.plot(plotx,normal_pdf(plotx,a,np.sqrt(b_sq)),'r')\n",
    "plt.title('Prior density')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p>Because we chose a conjugate prior-likelihood pair, we are able to compute the posterior analytically. From Bayes rule, we have:\n",
    "$$ p(\\mu|x_1,\\ldots,x_N,a,b^2,\\sigma^2) = \\frac{p(x_1,\\ldots,x_N|\\mu,\\sigma^2)p(\\mu|a,b^2)}{p(x_1,\\ldots,x_N|\\sigma^2,a,b^2)}$$\n",
    "The likelihood term can be factorised as:\n",
    "$$\n",
    "p(x_1,\\ldots,x_N|\\mu,\\sigma^2) = \\prod_{i=1}^N p(x_i|\\mu,\\sigma^2) = \\prod_{i=1}^N {\\cal N}(x_i|\\mu,\\sigma^2)\n",
    "$$\n",
    "Because the prior and likelihood are conjugate (both are Gaussian), we know that the posterior must be Gaussian. Given this, and the fact that the term in the denominator (the marginal likelihood) doesn't include $\\mu$ and can be ignored, we're left with matching the $\\mu$ terms in the posterior with those in the product of the likelihood and the prior. Ignoring the constant, the posterior can be written as:\n",
    "$$\n",
    "p(\\mu|x_1,\\ldots,x_N,a,b^2,\\sigma^2) = {\\cal N}(\\mu|c,d^2) \\propto \\exp\\left(-\\frac{1}{2d^2}(\\mu-c)^2\\right)\n",
    "$$\n",
    "The product of the prior and the likelihood is proportional to:\n",
    "$$\n",
    "\\exp\\left(-\\frac{1}{2b^2}(\\mu-a)^2 - \\frac{1}{2\\sigma^2}\\sum_{i=1}^N (x_i-\\mu)^2 \\right)\n",
    "$$\n",
    "To work out $c$ and $d^2$, we can match the $\\mu$ and $\\mu^2$ terms in the two expressions. Starting with $\\mu^2$ we have:\n",
    "$$\n",
    "\\mbox{Posterior}: -\\frac{1}{2d^2},~~~~\\mbox{Prior times likelihood}: -\\frac{1}{2b^2} - \\frac{N}{2\\sigma^2}\n",
    "$$\n",
    "Therefore:\n",
    "$$ \n",
    "-\\frac{1}{2d^2} = -\\frac{1}{2b^2} - \\frac{N}{2\\sigma^2}\n",
    "$$\n",
    "and\n",
    "$$\n",
    "d^2 = \\left(\\frac{1}{b^2} + \\frac{N}{\\sigma^2}\\right)^{-1}\n",
    "$$\n",
    "To find $c$ we equate the $\\mu$ terms:\n",
    "$$\n",
    "\\mbox{Posterior}: \\frac{c}{d^2},~~~~\\mbox{Prior times likelihood}: \\frac{a}{b^2} + \\frac{\\sum_{i=1}^N x_i}{\\sigma^2}\n",
    "$$\n",
    "Therefore:\n",
    "$$\n",
    "\\frac{c}{d^2} = \\frac{a}{b^2} + \\frac{\\sum_{i=1}^N x_i}{\\sigma^2}\n",
    "$$\n",
    "and\n",
    "$$\n",
    "c = d^2\\left(\\frac{a}{b^2} + \\frac{\\sum_{i=1}^N x_i}{\\sigma^2}\\right)\n",
    "$$\n",
    "<p>We now add the data one at a time, and see how the posterior density changes. In the plots, the red curve is the prior, the blue curve the posterior and the black circles the data. Remember that the posterior is over the mean value, so it may well not cover all of the data.</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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V39bSlq8UzVWvvw633ALz52fEKp7jj4cnnrBcl/OGDYOTToLbbkv6pX/7G/zm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\nTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1084761d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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9wsLCwm1f5+fnk5+fn+0QTKaMHg2//a31zitYssT926lThk5w++0wfDice65b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\nxsSEJXRjjIkJS+jGGBMT/w8S6obVgeZV2QAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1085938d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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avP12qCWYJC/3ra0F2qc8bpd8rkaKior2fF5QUEBBQUFND2WiZPduGD7c3Z7e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\nIyzQjTEmR1igG2NMjvh/vLQZ4ml9XV8AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1085c9d10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10869dd90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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XNm92f/pp92OPdW/Txv2ss9yffdb9iy9yrqeq9evdv/Md93PPDYePUn1rSppM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\nkhIKdBGRlFCgi4ikxP8DfHFXL8e4zEoAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1086d7710>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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EE92POMJ99eqIvomN+B6lQZSBPgx4NuPx5TV76cBdwCkZj+cCnWs5VkE+fFyu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\nAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1087be210>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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/94sucp86NbxeoJL8fBGK4/ONG+fevbv79OlNb7dq1Srfc889NwVx/T+DBg3y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\nCnQRkYz4/6fWrXtMuuvGAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1087ebf90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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eybl/Qe1WOjAROCHn/lygRx37Ksmbj8sll1wSbwHr14dEuPVW94sucj/6aPdu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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1088d5450>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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rfD/Gjh3bdAG1te6TJ7v/4hfuRx/tYzp2bPu+MlaudD/sMPczznBfssR9zJj8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\nkYRQoIuIJIQCXUQkIf4/LHKdROPqJ8kAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10890c390>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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WNO8zZtt3tvc3VVOpiTLQRwFP1Xl8Tf1ROnAPcH6dx7OBPg3sqyAfPi6TJk2K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\ntF4AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1089ec690>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "all_c = []\n",
    "all_d_sq = []\n",
    "for i in np.arange(x.size):\n",
    "    d_sq = 1.0/(1.0/b_sq + (i+1.0)/sig_sq)\n",
    "    c = d_sq*(a/b_sq + x[0:i+1].sum()/sig_sq)\n",
    "    all_c.append(c)\n",
    "    all_d_sq.append(d_sq)\n",
    "    plt.figure()\n",
    "    plt.plot(plotx,normal_pdf(plotx,a,np.sqrt(b_sq)),'r')\n",
    "    plt.plot(plotx,normal_pdf(plotx,c,np.sqrt(d_sq)),'b')\n",
    "    plt.plot(x[0:i+1],np.zeros_like(x[0:i+1]),'ko',markersize=10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p>We can also plot the evolution of the posterior mean ($c$) and variance ($d^2$).</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x10924e350>"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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PtGVmHYGrgD+GXUuixX55/U93nwHg7mXu/nXIZSVaQ+D02GoETYHtIddTL+6+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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x104465690>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(np.arange(x.size)+1,all_c,'k')\n",
    "plt.plot([1,x.size+1],[true_mu,true_mu],'k--')\n",
    "plt.xlabel('N')\n",
    "plt.ylabel('c')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x104437d90>"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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7ItYDvwFOL6KQrfy8nArcnD6/GZi2vc+pq/8czUzSASS/dSwotpKNp4oWAy8A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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1096b2c90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(np.arange(x.size)+1,all_d_sq,'r')\n",
    "plt.xlabel('N')\n",
    "plt.ylabel('$d^2$')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "    "
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.11"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}
